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Special Writing Assignment 1

Special Writing Assignment 1 - Problem 1 Work the following...

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Problem 1 Work the following problem, write it up carefully (on a separate sheet of paper from the homework), and turn it in on Friday, October 8: Problem : Prove that if x is rational and y is irrational, then x + y is irrational. Proof : We will use proof by contradiction. Since this is a conditional statement, P => Q, we assume P and not Q. That is, assume that x is rational, y is irrational, and x + y is rational. Since x is rational, it can be written as the quotient of two integers, p/q, where q neq 0. Also, x + y is rational means that it can be written as the quotient of two integers, r/s, where s neq 0. Then, x + y = p/q + y = r/s. So, y = r/s - p/q. Or, y = (rq - ps)/ qs. But, rq - ps is an integer, and so is qs. Also, qs neq 0. Therefore, y is the quotient of two integers and is thus rational. But, this is a contradiction to our premise that y is irrational. We thus conclude that P => Q, or that if x is rational and y is irrational, then x + y is irrational. qed.

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Special Writing Assignment 1 - Problem 1 Work the following...

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